In presence of external magnetic field, a gauge field appears in the Hofstadter Hamiltonian with the hopping terms which is known as Peierls substitution [1]}.
\(\mathcal {H}_{Hf} &=& \sum _{m} \left( t ^{2\frac{p}{q} m_2} a_{m_1 +1}^\dagger a_{m_1} + t a_{m_2 +1}^\dagger a_{m_2} + h.c.\right) \nonumber \\&+& \sum _m U a_{m}^\dagger a_{m}^\dagger a_m a_m + \sum _{<mn>} V_{|m-n|} a_{m}^\dagger a_{n}^\dagger a_n a_m\,.\) 
